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The companion equations and the moyal-nahm equations

Abstract

The first part of this thesis is concerned with the companion equations. These are equations of motion for the companion Lagrangian which is proposed to be the Lagrangian for a field theory associated with strings and branes, similar to the Klein-Gordon field description for particles. The form of this Lagrangian can be related to the Hamilton-Jacobi formalism for strings and branes. Some solutions to the companion equations are found and their integrability is discussed. There is an equivalence between the equations of motion for different companion Lagrangians when some constraints are applied. Under these constraints, the companion equations for a Lagrangian without a square root are equivalent to the companion equations for a Lagrangian with a square root but in one dimension less. The appearance of Universal Field Equations, generalised Bateman equations, in the companion equations leads to the study of an iterative procedure for Lagrangians which are homogeneous of weight one in the first derivatives in the fields the theory describes. The Universal Field equations appear after several iterations. Also, it is shown how Lagrangians for a large family of field theories are a divergence or vanish on the space of solutions of the equations of motion. Such theories could be called 'pseudo-topological'.The second part of this thesis is concerned with finding solutions to the Moyal-Nahm equations in four and eight dimensions. These equations are the Nahm equations, which give a set of solutions to self-dual Yang-Mills, but with the commutators replaced with Moyal brackets. Solutions are found in terms of generalised Wigner functions. Also, matrix representations of the algebra generated by the equivalent Nahm equations in eight dimensions are obtained. Solutions to the Nahm equations in eight dimensions are also given.